What is a permutation?

A permutation is an arrangement of items where the order matters. P(n, r) counts how many different ordered sequences of r items can be formed by choosing from n distinct items.

What is the formula for permutations?

P(n, r) = n! / (n - r)!. For example, P(5, 2) = 5! / 3! = 5 × 4 = 20 — there are 20 ways to arrange 2 items chosen from a set of 5.

What's the difference between a permutation and a combination?

Permutations count arrangements where order matters (e.g., 1st, 2nd, 3rd place in a race). Combinations count selections where order doesn't matter (e.g., choosing a committee of 3 people). Use the Combination Calculator when order doesn't matter.

What happens if r is greater than n?

P(n, r) is undefined when r > n, since you can't arrange more items than exist in the set. The calculator will show an error in this case.

When r equals n, P(n, n) = n!, since you're arranging every item in the set in every possible order.

Yes. P(n, 0) always equals 1 (there's exactly one way to arrange zero items — do nothing), and P(0, 0) also equals 1.

Where are permutations used in real life?

Permutations are used to count possible race finishes, password combinations where order matters, seating arrangements, and any scenario where you need to know how many distinct orderings are possible.

Can this calculator handle large values of n?

Yes, the calculator uses exact integer arithmetic that avoids computing oversized intermediate factorials, so it stays accurate for values of n up to 1,000.

Permutation Calculator — Find P(n, r) With Step-by-Step Solutions

Our Permutation Calculator finds P(n, r), the number of ordered arrangements of r items chosen from a set of n distinct items. Enter your values for n and r and get the exact result instantly, along with a step-by-step breakdown of the formula n! / (n - r)! so you can see exactly how the answer was derived — useful for checking homework, planning schedules, or working through probability problems.

Quick Answer

A permutation counts ordered arrangements: P(n, r) = n! / (n - r)! gives the number of ways to arrange r items chosen from n distinct items where order matters. Enter n and r below for an instant result with step-by-step working.

Total items (n)

Items arranged (r)

Enter n and r, then click Calculate.

How to Use the Permutation Calculator — Calculate P(n,r) Online

1
Enter the total number of items, n, into the first field.
2
Enter the number of items being arranged, r, into the second field.
3
Click 'Calculate' to find P(n, r) — the number of ways to order r items from n total.
4
Review the step-by-step breakdown showing the formula n! / (n - r)! applied to your values.
5
Click 'Copy' to copy the result to your clipboard.

Why Use Permutation Calculator — Calculate P(n,r) Online?

Permutations answer 'in how many orders can this happen?' — a question that comes up constantly in probability, scheduling, password analysis, and competition math. The formula n! / (n - r)! is simple to state but easy to compute wrong by hand once n grows past single digits, since it involves large factorials that partially cancel. Our calculator computes the result directly using an efficient method that avoids ever forming the full, oversized factorial, so it stays fast and exact even for large values of n and r.

Permutation vs Combination

	Permutation P(n,r)	Combination C(n,r)
Does order matter?	Yes — different orders count separately	No — only the group of items counts
Formula	n! / (n − r)!	n! / (r!(n − r)!)
Example: n=5, r=2	P(5,2) = 20	C(5,2) = 10
Typical use case	Race rankings, PIN codes, seating order	Lottery numbers, committees, card hands

Frequently Asked Questions

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